Basic invariants
Galois action
Roots of defining polynomial
The roots of $f$ are computed in $\Q_{ 127 }$ to precision 9.
Roots:
| $r_{ 1 }$ |
$=$ |
$ 3 + 62\cdot 127 + 121\cdot 127^{2} + 107\cdot 127^{3} + 125\cdot 127^{4} + 2\cdot 127^{5} + 54\cdot 127^{6} + 42\cdot 127^{7} + 107\cdot 127^{8} +O\left(127^{ 9 }\right)$ |
| $r_{ 2 }$ |
$=$ |
$ 12 + 37\cdot 127 + 86\cdot 127^{2} + 75\cdot 127^{3} + 112\cdot 127^{4} + 109\cdot 127^{5} + 55\cdot 127^{6} + 33\cdot 127^{7} + 83\cdot 127^{8} +O\left(127^{ 9 }\right)$ |
| $r_{ 3 }$ |
$=$ |
$ 29 + 68\cdot 127 + 25\cdot 127^{2} + 84\cdot 127^{3} + 42\cdot 127^{4} + 87\cdot 127^{5} + 101\cdot 127^{6} + 48\cdot 127^{7} + 126\cdot 127^{8} +O\left(127^{ 9 }\right)$ |
| $r_{ 4 }$ |
$=$ |
$ 54 + 120\cdot 127 + 35\cdot 127^{2} + 74\cdot 127^{3} + 115\cdot 127^{4} + 85\cdot 127^{5} + 52\cdot 127^{6} + 40\cdot 127^{7} + 113\cdot 127^{8} +O\left(127^{ 9 }\right)$ |
| $r_{ 5 }$ |
$=$ |
$ 74 + 6\cdot 127 + 91\cdot 127^{2} + 52\cdot 127^{3} + 11\cdot 127^{4} + 41\cdot 127^{5} + 74\cdot 127^{6} + 86\cdot 127^{7} + 13\cdot 127^{8} +O\left(127^{ 9 }\right)$ |
| $r_{ 6 }$ |
$=$ |
$ 99 + 58\cdot 127 + 101\cdot 127^{2} + 42\cdot 127^{3} + 84\cdot 127^{4} + 39\cdot 127^{5} + 25\cdot 127^{6} + 78\cdot 127^{7} +O\left(127^{ 9 }\right)$ |
| $r_{ 7 }$ |
$=$ |
$ 116 + 89\cdot 127 + 40\cdot 127^{2} + 51\cdot 127^{3} + 14\cdot 127^{4} + 17\cdot 127^{5} + 71\cdot 127^{6} + 93\cdot 127^{7} + 43\cdot 127^{8} +O\left(127^{ 9 }\right)$ |
| $r_{ 8 }$ |
$=$ |
$ 125 + 64\cdot 127 + 5\cdot 127^{2} + 19\cdot 127^{3} + 127^{4} + 124\cdot 127^{5} + 72\cdot 127^{6} + 84\cdot 127^{7} + 19\cdot 127^{8} +O\left(127^{ 9 }\right)$ |
Generators of the action on the roots
$r_1, \ldots, r_{ 8 }$
| Cycle notation |
| $(1,8)(2,7)(3,6)(4,5)$ |
| $(2,7)(3,4,6,5)$ |
| $(1,4)(2,3)(5,8)(6,7)$ |
| $(3,6)(4,5)$ |
| $(1,7)(2,8)(3,5)(4,6)$ |
Character values on conjugacy classes
| Size | Order | Action on
$r_1, \ldots, r_{ 8 }$
| Character value |
| $1$ | $1$ | $()$ | $4$ |
| $1$ | $2$ | $(1,8)(2,7)(3,6)(4,5)$ | $-4$ |
| $2$ | $2$ | $(1,7)(2,8)(3,5)(4,6)$ | $0$ |
| $2$ | $2$ | $(1,2)(3,5)(4,6)(7,8)$ | $0$ |
| $2$ | $2$ | $(3,6)(4,5)$ | $0$ |
| $4$ | $2$ | $(1,4)(2,3)(5,8)(6,7)$ | $0$ |
| $4$ | $4$ | $(1,6,2,4)(3,7,5,8)$ | $0$ |
| $4$ | $4$ | $(1,4,2,6)(3,8,5,7)$ | $0$ |
| $4$ | $4$ | $(2,7)(3,4,6,5)$ | $0$ |
| $4$ | $4$ | $(2,7)(3,5,6,4)$ | $0$ |
| $4$ | $4$ | $(1,6,8,3)(2,5,7,4)$ | $0$ |
The blue line marks the conjugacy class containing complex conjugation.
